Antidifferentiation

Antidifferentiation is the process of, given a function \(f(x)\), finding another function \(F(x)\) whose derivative is the given function, \(\frac{d}{dx} F(x) = F^\prime(x) = f(x)\). Antidifferentiation and the antiderivative are part of a larger process called integration. Formally, the antiderivative of \(f(x)\) is the set of functions \(F(x) + C\) such that \[\frac{d}{dx}[F(x) + C] = f(x)\] where \(C\) is what’s called the constant of integration. This can be restated as follows: if two functions \(F(x)\) and \(G(x)\) have the same derivative \(f(x)\), then \(F(x)\) and \(G(x)\) differ by at most a constant \(F(x) = G(x) + C\). …

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Economics Applications of Derivatives

Some interesting things you can do with derivatives in the context of economics are to: Find the price elasticity of a demand function. Find the maximum of a total-revenue function. Characterize demand in terms of elasticity. Retailers and manufacturers often need to know how a small change in price will affect the demand of a product. If a small increase in price produces no change in demand, then the price increase is a reasonable decision to make. …

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Derivatives of Exponential and Logarithmic Functions

The derivative of an exponential function of the form \(a^x\) is \[\frac{d}{dx}a^x = (\ln a)a^x\] from sympy import * from sympy.abc import x expr = 2**x expr2 = 3**(2*x) diff(expr) ## 2**x*log(2) diff(expr2) ## 2*3**(2*x)*log(3) The derivative of a function of the form \(\log_ax\)is \[\frac{d}{dx}\log_a x = \frac{1}{\ln a} \cdot \frac{1}{x}\] expr3 = log(x, 8) expr4 = log((x**2 + 1), 3) diff(expr3) ## 1/(x*log(8)) diff(expr4) ## 2*x/((x**2 + 1)*log(3))

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Exponential Decay

In the equation of population growth, \(dP/dt = kP\), the constant \(k\) is given by \(k = (\text{birth rate}) - (\text{death rate})\). Thus, a population grows only when the birth rate is greater than the death rate. When the death rate is greater than the birth rate we have a decrease or “decay”. Our equation for population decay becomes \(dP/dt = -kP\), where \(k >0\). The function that satisfies that rate of return is then …

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Exponential Functions Business Applications

We go over some textbook exercises for exponential functions. Franchise Expansion A franchise business is selling franchises for pizza shops throughout the country. The marketing manager estimates that the number of franchises, \(N\), will increase at the rate of 10% per year, that is, \[\frac{dN}{dt} = 0.10N\] Find the function that satisfies this equation, assuming that the number of franchises at \(t = 0\) is 50. How many franchises will be there in 20 years? …

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